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New self-dual codes of length 68 from a $ 2 \times 2 $ block matrix construction and group rings
1. | Department of Algebra, Uzhgorod National University, Uzhgorod, Ukraine |
2. | Department of Mathematical and Physical Sciences, University of Chester, UK |
3. | Harmony School of Technology, Houston, TX 77038, USA |
Many generator matrices for constructing extremal binary self-dual codes of different lengths have the form $ G = (I_n \ | \ A), $ where $ I_n $ is the $ n \times n $ identity matrix and $ A $ is the $ n \times n $ matrix fully determined by the first row. In this work, we define a generator matrix in which $ A $ is a block matrix, where the blocks come from group rings and also, $ A $ is not fully determined by the elements appearing in the first row. By applying our construction over $ \mathbb{F}_2+u\mathbb{F}_2 $ and by employing the extension method for codes, we were able to construct new extremal binary self-dual codes of length 68. Additionally, by employing a generalised neighbour method to the codes obtained, we were able to construct many new binary self-dual $ [68, 34, 12] $-codes with the rare parameters $ \gamma = 7, 8 $ and $ 9 $ in $ W_{68, 2}. $ In particular, we find 92 new binary self-dual $ [68, 34, 12] $-codes.
References:
[1] |
W. Bosma, J. Cannon and C. Playoust,
The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235-265.
doi: 10.1006/jsco.1996.0125. |
[2] |
S. Buyuklieva and I. Boukliev,
Extremal self-dual codes with an automorphism of order 2, IEEE Trans. Inform. Theory, 44 (1998), 323-328.
doi: 10.1109/18.651059. |
[3] |
J. H. Conway and N. J. A. Sloane,
A new upper bound on the minimal distance of self-dual codes, IEEE Trans. Inform. Theory, 36 (1990), 1319-1333.
doi: 10.1109/18.59931. |
[4] |
S. T. Dougherty, P. Gaborit, M. Harada and P. Sole,
Type II codes over $\mathbb{F}_2+u\mathbb{F}_2$, IEEE Trans. Inform. Theory, 45 (1999), 32-45.
doi: 10.1109/18.746770. |
[5] |
S. T. Dougherty, J. Gildea and A. Kaya, Quadruple bordered constructions of self-dual codes from group rings over Frobenius rings, Cryptogr. Commun., (2019).
doi: 10.1007/s12095-019-00380-8. |
[6] |
S. T. Dougherty, J. Gildea, A. Korban and A. Kaya, Composite constructions of self-dual codes from group rings and new extremal self-dual binary codes of length 68, Adv. Math. Comm., (2019).
doi: 10.1016/j.ffa.2020.101692. |
[7] |
S. T. Dougherty, J. Gildea, A. Korban, A. Kaya, A. Tylshchak and B. Yildiz,
Bordered constructions of self-dual codes from group rings, Finite Fields Appl., 57 (2019), 108-127.
doi: 10.1016/j.ffa.2019.02.004. |
[8] |
S. T. Dougherty, J. Gildea, R. Taylor and A. Tylshchak,
Group rings, g-codes and constructions of self-dual and formally self-dual codes, Des., Codes and Cryptog., Designs, 86 (2018), 2115-2138.
doi: 10.1007/s10623-017-0440-7. |
[9] |
S. T. Dougherty, S. Karadeniz and B. Yildiz,
Codes over $R_k$, gray maps and their binary images, Finite Fields Appl., 17 (2011), 205-219.
doi: 10.1016/j.ffa.2010.11.002. |
[10] |
S. T. Dougherty, J. L. Kim, H. Kulosman and H. Liu,
Self-dual codes over commutative Frobenius rings, Finite Fields Appl., 16 (2010), 14-26.
doi: 10.1016/j.ffa.2009.11.004. |
[11] |
J. Gildea, A. Kaya, A. Korban and B. Yildiz, Constructing self-dual codes from group rings and reverse circulant matrices, Adv. Math. Comm..
doi: 10.3934/amc.2020077. |
[12] |
J. Gildea, A. Kaya, A. Korban and B. Yildiz, New extremal binary self-dual codes of length 68 from generalized neighbours, Finite Fields Appl., (2020).
doi: 10.1016/j.ffa.2020.101727. |
[13] |
J. Gildea, A. Kaya, R. Taylor and B. Yildiz,
Constructions for self-dual codes induced from group rings, Finite Fields Appl., 51 (2018), 71-92.
doi: 10.1016/j.ffa.2018.01.002. |
[14] |
M. Harada and A. Munemasa,
Some restrictions on weight enumerators of singly even self-dual codes, IEEE Trans. Inform. Theory, 52 (2006), 1266-1269.
doi: 10.1109/TIT.2005.864416. |
[15] |
T. Hurley,
"Group Rings and Rings of Matrices", J. Pure Appl. Math., 31 (2006), 319-335.
|
[16] |
S. Karadeniz, B. Yildiz and N. Aydin,
Extremal binary self-dual codes of lengths 64 and 66 from four-circulant constructions over $\mathbb{F}_2+u\mathbb{F}_2$, Filomat, 28 (2014), 937-945.
doi: 10.2298/FIL1405937K. |
[17] |
A. Kaya,
New extremal binary self-dual codes of lengths 64 and 66 from $R_{2}$-lifts, Finite Fields Appl., 46 (2017), 271-279.
doi: 10.1016/j.ffa.2017.04.003. |
[18] |
A. Kaya and B. Yildiz,
Various constructions for self-dual codes over rings and new binary self-dual codes, Discrete Math., 339 (2016), 460-469.
doi: 10.1016/j.disc.2015.09.010. |
[19] |
E. M. Rains,
Shadow bounds for self-dual codes, IEEE Trans. Inf. Theory, 44 (1998), 134-139.
doi: 10.1109/18.651000. |
[20] |
N. Yankov, M. H. Lee, M. Gurel and M. Ivanova,
Self-dual codes with an automorphism of order 11, IEEE Trans. Inform. Theory, 61 (2015), 1188-1193.
doi: 10.1109/TIT.2015.2396915. |
[21] |
N. Yankov, M. Ivanova and M. H. Lee,
Self-dual codes with an automorphism of order 7 and s-extremal codes of length 68, Finite Fields Appl., 51 (2018), 17-30.
doi: 10.1016/j.ffa.2017.12.001. |
[22] |
N. Yankov and D. Anev, On the self-dual codes with an automorphism of order 5, AAECC, (2019).
doi: 10.1007/s00200-019-00403-0. |
show all references
References:
[1] |
W. Bosma, J. Cannon and C. Playoust,
The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235-265.
doi: 10.1006/jsco.1996.0125. |
[2] |
S. Buyuklieva and I. Boukliev,
Extremal self-dual codes with an automorphism of order 2, IEEE Trans. Inform. Theory, 44 (1998), 323-328.
doi: 10.1109/18.651059. |
[3] |
J. H. Conway and N. J. A. Sloane,
A new upper bound on the minimal distance of self-dual codes, IEEE Trans. Inform. Theory, 36 (1990), 1319-1333.
doi: 10.1109/18.59931. |
[4] |
S. T. Dougherty, P. Gaborit, M. Harada and P. Sole,
Type II codes over $\mathbb{F}_2+u\mathbb{F}_2$, IEEE Trans. Inform. Theory, 45 (1999), 32-45.
doi: 10.1109/18.746770. |
[5] |
S. T. Dougherty, J. Gildea and A. Kaya, Quadruple bordered constructions of self-dual codes from group rings over Frobenius rings, Cryptogr. Commun., (2019).
doi: 10.1007/s12095-019-00380-8. |
[6] |
S. T. Dougherty, J. Gildea, A. Korban and A. Kaya, Composite constructions of self-dual codes from group rings and new extremal self-dual binary codes of length 68, Adv. Math. Comm., (2019).
doi: 10.1016/j.ffa.2020.101692. |
[7] |
S. T. Dougherty, J. Gildea, A. Korban, A. Kaya, A. Tylshchak and B. Yildiz,
Bordered constructions of self-dual codes from group rings, Finite Fields Appl., 57 (2019), 108-127.
doi: 10.1016/j.ffa.2019.02.004. |
[8] |
S. T. Dougherty, J. Gildea, R. Taylor and A. Tylshchak,
Group rings, g-codes and constructions of self-dual and formally self-dual codes, Des., Codes and Cryptog., Designs, 86 (2018), 2115-2138.
doi: 10.1007/s10623-017-0440-7. |
[9] |
S. T. Dougherty, S. Karadeniz and B. Yildiz,
Codes over $R_k$, gray maps and their binary images, Finite Fields Appl., 17 (2011), 205-219.
doi: 10.1016/j.ffa.2010.11.002. |
[10] |
S. T. Dougherty, J. L. Kim, H. Kulosman and H. Liu,
Self-dual codes over commutative Frobenius rings, Finite Fields Appl., 16 (2010), 14-26.
doi: 10.1016/j.ffa.2009.11.004. |
[11] |
J. Gildea, A. Kaya, A. Korban and B. Yildiz, Constructing self-dual codes from group rings and reverse circulant matrices, Adv. Math. Comm..
doi: 10.3934/amc.2020077. |
[12] |
J. Gildea, A. Kaya, A. Korban and B. Yildiz, New extremal binary self-dual codes of length 68 from generalized neighbours, Finite Fields Appl., (2020).
doi: 10.1016/j.ffa.2020.101727. |
[13] |
J. Gildea, A. Kaya, R. Taylor and B. Yildiz,
Constructions for self-dual codes induced from group rings, Finite Fields Appl., 51 (2018), 71-92.
doi: 10.1016/j.ffa.2018.01.002. |
[14] |
M. Harada and A. Munemasa,
Some restrictions on weight enumerators of singly even self-dual codes, IEEE Trans. Inform. Theory, 52 (2006), 1266-1269.
doi: 10.1109/TIT.2005.864416. |
[15] |
T. Hurley,
"Group Rings and Rings of Matrices", J. Pure Appl. Math., 31 (2006), 319-335.
|
[16] |
S. Karadeniz, B. Yildiz and N. Aydin,
Extremal binary self-dual codes of lengths 64 and 66 from four-circulant constructions over $\mathbb{F}_2+u\mathbb{F}_2$, Filomat, 28 (2014), 937-945.
doi: 10.2298/FIL1405937K. |
[17] |
A. Kaya,
New extremal binary self-dual codes of lengths 64 and 66 from $R_{2}$-lifts, Finite Fields Appl., 46 (2017), 271-279.
doi: 10.1016/j.ffa.2017.04.003. |
[18] |
A. Kaya and B. Yildiz,
Various constructions for self-dual codes over rings and new binary self-dual codes, Discrete Math., 339 (2016), 460-469.
doi: 10.1016/j.disc.2015.09.010. |
[19] |
E. M. Rains,
Shadow bounds for self-dual codes, IEEE Trans. Inf. Theory, 44 (1998), 134-139.
doi: 10.1109/18.651000. |
[20] |
N. Yankov, M. H. Lee, M. Gurel and M. Ivanova,
Self-dual codes with an automorphism of order 11, IEEE Trans. Inform. Theory, 61 (2015), 1188-1193.
doi: 10.1109/TIT.2015.2396915. |
[21] |
N. Yankov, M. Ivanova and M. H. Lee,
Self-dual codes with an automorphism of order 7 and s-extremal codes of length 68, Finite Fields Appl., 51 (2018), 17-30.
doi: 10.1016/j.ffa.2017.12.001. |
[22] |
N. Yankov and D. Anev, On the self-dual codes with an automorphism of order 5, AAECC, (2019).
doi: 10.1007/s00200-019-00403-0. |
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